TITLE: What is the most elegant and simple proof for the law of cosines?
QUESTION [7 upvotes]: Given 2 sides, and an angle between those two sides, what is the simplest proof you can come up with to find the measure of the 3rd side?

REPLY [2 votes]: Here's a proof similar to the dot product proof in previous answers, but using complex numbers and Euler's formula:
$$
  |a-b|^2 = (a-b) \overline{(a-b)} = a \overline{a} + b \overline{b} - (\overline{a} b + a \overline{b}) = \\
|a|^2 + |b|^2 - |a||b|(e^{i(\arg{b} - \arg{a})} + e^{i(\arg{a} - \arg{b})}) = \\  |a|^2 + |b|^2 - 2|a||b|\cos{\theta}
$$